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Theorem 0nelopabOLD 5524
Description: Obsolete version of 0nelopab 5523 as of 3-Oct-2024. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
0nelopabOLD ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem 0nelopabOLD
StepHypRef Expression
1 elopab 5483 . . 3 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 nfopab1 5174 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfel2 2924 . . . . 5 𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43nfn 1861 . . . 4 𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 5175 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
65nfel2 2924 . . . . . 6 𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
76nfn 1861 . . . . 5 𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
8 vex 3448 . . . . . . . 8 𝑥 ∈ V
9 vex 3448 . . . . . . . 8 𝑦 ∈ V
108, 9opnzi 5430 . . . . . . 7 𝑥, 𝑦⟩ ≠ ∅
11 nesym 2999 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩)
12 pm2.21 123 . . . . . . . 8 (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1311, 12sylbi 216 . . . . . . 7 (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
1410, 13ax-mp 5 . . . . . 6 (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
1514adantr 482 . . . . 5 ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
167, 15exlimi 2211 . . . 4 (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
174, 16exlimi 2211 . . 3 (∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
181, 17sylbi 216 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 id 22 . 2 (¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2018, 19pm2.61i 182 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2942  c0 4281  cop 4591  {copab 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167
This theorem is referenced by: (None)
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